Rectified 7-orthoplexes

Rectified 7-orthoplex

The rectified 7-orthoplex is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb. : or

Alternate names

• rectified heptacross
• rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon

Images

Construction

There are two Coxeter groups associated with the rectified heptacross, one with the C7 or [4,3,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D7 or [34,1,1] Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified heptacross, centered at the origin, edge length \sqrt{2}\ are all permutations of: : (±1,±1,0,0,0,0,0)

Root vectors

Its 84 vertices represent the root vectors of the simple Lie group D7. The vertices can be seen in 3 hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B7 and C7 simple Lie groups.

Birectified 7-orthoplex

Alternate names

• Birectified heptacross
• Birectified hecatonicosoctaexon (Acronym barz) (Jonathan Bowers) - birectified 128-faceted polyexon

Images

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length \sqrt{2}\ are all permutations of: : (±1,±1,±1,0,0,0,0)

Trirectified 7-orthoplex

A trirectified 7-orthoplex is the same as a trirectified 7-cube.

Notes

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com,
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• o3x3o3o3o3o4o - rez, o3o3x3o3o3o4o - barz

References

1. Klitzing, (o3o3x3o3o3o4o - rez)
2. Klitzing, (o3o3x3o3o3o4o - barz)